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arXiv · 1012.2641

Nordhaus-Gaddum-type theorem for rainbow connection number of graphs

Lily Chen·Xueliang Li·Huishu Lian
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Abstract

An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of $G$, denoted $rc(G)$, is the minimum number of colors that are used to make $G$ rainbow connected. In this paper we give a Nordhaus-Gaddum-type result for the rainbow connection number. We prove that if $G$ and $\bar{G}$ are both connected, then $4\leq rc(G)+rc(\bar{G})\leq n+2$. Examples are given to show that the upper bound is sharp for all $n\geq 4$, and the lower bound is sharp for all $n\geq 8$. For the rest small $n=4,5,6,7,$ we also give the sharp bounds.

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Lily Chen, Xueliang Li, Huishu Lian. 2010-12-14. Nordhaus-Gaddum-type theorem for rainbow connection number of graphs. https://arxiv.org/abs/1012.2641

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